Question

The accompanying figure shows a network of one-way streets with traffic flowing in the directions indicated. The flow rates along the streets are measured as the average number of vehicles per hour. 300 800 275 300 700 BI 200 250 25 (a) Set up a linear system whose solution provides the unknown flow rates. С X1- X2 - 25 x2 x3 --500 x3 + x4 = 450 +x4 = 25 С X1.x2 - 25 --500 x2X3 x3 +x4 = 450 + X425 = 450 X3 · X4 = .500 = 25 + x4 = 25 - 25 *2 - X3 x3 - X4 =-500 -450 + X425 (b) Solve the system for the unknown flow rates. If the system has infintely many solution, consider X4 s. (c) If the flow along the road from A to B must be reduced for construction, what is the minimum flow that is required to keep traffic flowing on all roads? The minimum required flow vehicles per hour.

Answer 1

(a) The correct linear system is following:

x2-x3 =25 2 X3 X4 500 X1 -x2 -x1 + X4 = 25

The flow balance equations at each node are following:

x1+250=x2+700

x2+275=x3+300

x3+800=x4+300

x4+200=x1+225

Simplifying these equations, we get

x1 - x2 = 450

x2 - x3 = 25

x3 - x4 = -500

x4 - x1 = 25

(b)

x4 = s,

x1 = s - 25

x2 = s - 25 - 450 = s - 475

x3 = s - 475 - 25 = s - 500

(c)

In the above solution, we see that x3 = s - 500

For non-negative flow, value of s must be >= 500

Flow from A to B is represented by x4, which is s

Therefore, Minimum required flow = 500 vehicles per hour